Methods for Ellipsometry Determination of Sample Characteristics

Abstract

The present invention relates to a method for determining the characteristics of a sample (12) by ellipsometric determination, the method comprising the following steps: - positioning the sample (12) in an ellipsometer (10), wherein the surface normal (n) of a measurement area of ​​the sample surface is tilted relative to a reference axis (z) of the ellipsometer (10), - measuring a Mueller matrix for the measurement area; - establishing at least a non-underdetermined set of equations by making the measured Mueller matrix equal to the matrix product of the following matrices: + a rotation matrix for an input rotation angle (γ), + an isotropic Mueller matrix in normalized NCS form, and + a rotation matrix for an output rotation angle (‑δ); - solving the set of equations according to parameters representing the sample characteristics to be determined. The present invention is characterized in that, when establishing the set of equations, the input rotation angle (γ) and the output rotation angle (‑δ) are assumed to be independent parameters.

Description

Technical Field

[0001] The present invention relates to a method for determining sample characteristics by ellipsometry, the method comprising the following steps:

[0002] - positioning the sample in an ellipsometer, wherein the surface normal of a measurement area of ​​the sample surface is tilted relative to a reference axis of the ellipsometer,

[0003] -Measure the Mueller matrix for the measurement area,

[0004] - establishing an at least non-underdetermined system of equations based on equating the measured Mueller matrix with the matrix product of,

[0005] + the rotation matrix about the input rotation angle,

[0006] + the isotropic Mueller matrix in normalized NCS form, and

[0007] + the rotation matrix about the output rotation angle,

[0008] - Solving a system of equations according to the parameters representing the characteristics of the sample to be determined. Background Art

[0009] This method is known from: W. Li et.al.: Characterization of curved surface layer by Mueller matrix ellipsometry", Journal of Vacuum Science & Technology B34, 020602 (2016); http: / / dx.doi.org / 10.1116 / 1.4943952 (W. Li et al.: "Characterization of curved surface layer by Mueller matrix ellipsometry", Journal of Vacuum Science & Technology B34, 020602 (2016); http: / / dx.doi.org / 10.1116 / 1.4943952).

[0010] Ellipsometry, in particular image-generating ellipsometry, is a long-established method for determining optical and geometrical properties of samples, in particular thin layers. For example, the determination of the thickness and / or the refractive index of a coating is a typical subject of ellipsometry tests, which can be used, for example, in the production of hardened and tempered surfaces, for example within the scope of quality control measures.

[0011] The determination of sample properties is based on the analysis of the angle-dependent reflection behavior of the sample. In this case, electromagnetic radiation with a defined polarization state, for example a laser beam which is broadened if necessary, is directed onto the sample, i.e. onto a sub-region of its surface which is referred to here as the measurement region, and the change in the polarization state of the electromagnetic radiation is measured after interaction with the sample or reflection on the sample. Technically relevant sample properties which can be determined with high precision using this method are, for example, the layer thickness of at least semi-transparent layers or layer systems, or optical or dielectric constants of almost any material.

[0012] Ellipsometry is an indirect measurement method, the evaluation of which is based on model optimization. The different measured variables according to the method principle are converted into ellipsometric transfer variables Ψ (Psi, amplitude information) and Δ (Delta, phase information), from which the sample property of interest is determined via parameterized fitting, i.e. by numerical error minimization as the parameters of the sample model are varied.

[0013] The basic principle of ellipsometry and in particular the structure of measuring instruments based on this principle (commonly referred to as ellipsometers) are known to those skilled in the art. Reference is made to DIN 50989-1:2018-01, which also specifies the general and commonly used terminology used here, in particular with regard to the Mueller representation on which it is based.

[0014] A special form of ellipsometry is the so-called imaging ellipsometry. Here, the illuminated measurement area is imaged onto a two-dimensionally resolved detector by means of imaging optics, so that a non-imaged ellipsometry measurement method can be performed simultaneously for each point of the measurement area, in which each detector pixel is used as a non-imaged detector for the respectively assigned measurement area point, and the effect of providing an image is obtained by an orderly overview of the detector pixels. Alternatively, within the scope of imaging ellipsometry, illumination that scans the sample surface can also be used. In this case, the detection is either carried out in a manner without providing an image and then orderly combining the point-by-point measurement results to form a so-called map (also referred to as "stitching"), or the detection is carried out in a manner that directly provides an image, that is, with the aid of a two-dimensionally resolved detector.

[0015] Ellipsometry measurements are always associated with specific angle settings. This is especially the setting of the nominal angle of incidence of the illumination light, the nominal angle of incidence of the probe light, and the polarization angles at the polarizer and analyzer. The nominal angle of incidence or angle of incidence is understood to be the angular position of the corresponding light relative to the reference axis of the ellipsometer. The reference axis is the axis to which the ellipsometer is calibrated. Typically, the reference axis is selected such that it coincides with the surface normal of a planar sample surface. This case, i.e. a planar sample, is also a typical application case and the most familiar one in the prior art.

[0016] In practice, however, the person skilled in the art is often faced with the task of measuring curved surfaces ellipsometrically. Merely by way of example, the quality control of a tempered coating on a (possibly aspherical) curved lens is mentioned here. For this purpose, the sample is usually measured step by step, respectively in sections, wherein the sample is repositioned before each measurement so that its surface normal in the measurement area coincides with the reference axis of the ellipsometer.

[0017] For example, US Pat. No. 7,800,756 B2 describes a method in which a sample is positioned with suitable equipment in such a way that successive positions of the sample are correctly oriented relative to a scanning ellipsometer, ie with the surface normal oriented coincidentally with the reference axis of the ellipsometer.

[0018] US Pat. No. 6,741,353 B1 describes another method in which the sample is likewise positioned by means of a motor so that the individual points on the surface are correctly oriented one after the other. An additional restriction here is that the object to be measured must have at least one axis of symmetry.

[0019] These methods are expensive in terms of equipment and slow to use.

[0020] US 7,768,660 B1 describes a method for measuring the coating on the inside of a cylinder. However, here the sample is oriented from the outset in such a way that the non-curved axis of the cylinder is parallel to the beam propagation direction. The influence of the second curved cylinder axis is neglected, since the radius of curvature is assumed to be large relative to the measurement area.

[0021] The generic publication mentioned at the outset discloses a less apparatus-based assumption, but rather a mathematical, evaluation-related assumption, in order to enable ellipsometric determination of curved surfaces. The corresponding method is referred to here as the “Li method”.

[0022] In Li's method, the sample to be measured is positioned in the ellipsometer, wherein at least a section of the sample surface is not correctly oriented in the above sense. A further consideration focuses on measurement regions that are tilted relative to a reference axis.

[0023] The so-called Mueller matrix is ​​measured for such a measurement area. The method behind the abbreviated expression "measurement of the Mueller matrix" is known to the person skilled in the art and is described, for example, in Tompkins, HG, Irene, EA: Handbook of ellipsometry; William Andrew Pub.; Springer, Norwich, NY; Heidelberg, Germany (2005). In general, it is described that a plurality of measurement processes of a sample are carried out in an ellipsometer, each with different settings for the nominal polarization angle. The individual measurement results are used to construct a 4×4 matrix in a predetermined manner. The resulting 4×4 matrix is ​​referred to here as the "measured Mueller matrix". This term is to be understood broadly, since, for reasons explained in more detail below, at least within the scope of the present invention, knowledge of the complete Mueller matrix, ie the measurement and structure, is not required, but rather a partial knowledge of the Mueller matrix is ​​sufficient, provided that at least five matrix elements are assigned values.

[0024] The measured Mueller matrix is ​​then used to set up the system of equations in such a way that the measured Mueller matrix is ​​equal to the matrix product explained below. The resulting system of equations (which includes up to 16 individual equations) is derived from the known matrix multiplication rules. The background of this measure is the assumption that the measured Mueller matrix corresponds to an ideal, i.e. isotropic, normalized and sample-characterizing Mueller matrix, which has undergone a coordinate transformation, in particular an input and output rotation around the same rotation angle α (excluding its sign) in the context of the (unoriented) measurement. In terms of mathematical expression, Li's method follows the following assumption:

[0025]

[0026] in, and The Mueller matrix represents the coordinate transformation (from the coordinate system of the ellipsometer to the eigensystem of the sample and vice versa) of the above input and output sides. Δ,Ψ is the Mueller matrix of the isotropic sample in the eigensystem, M′ Δ,Ψ is the measured Mueller matrix.

[0027] In this context, the term "eigensystem of the sample" refers to the input-side and output-side Cartesian coordinate systems defined by the respective p and s vectors (parallel or perpendicular polarization components) and the respective propagation directions of the incident and reflected light beams onto the sample. Similarly, when the sample normal corresponds to the (correct) orientation in which the ellipsometer is calibrated, then the "eigensystem of the ellipsometer" corresponds to the eigensystem of the sample.

[0028] Normalization of isotropic samples (that is, element m 11 The design of the Mueller matrix of ≡1) is characterized by the ellipsometric transfer parameters Δ and Ψ and is described in the eigensystem of the sample, as is known from the specialist literature:

[0029]

[0030] Among them, the parameters N, C, and S are defined as

[0031]

[0032] According to Li's method, as described above, the forward and backward (unknown) angle α rotations are used to transform the coordinates of the isotropic Mueller matrix. The rotation matrix in the Mueller representation generally has the following form

[0033]

[0034] As is well known to those skilled in the art.

[0035] In other words, Li's method follows the following premise:

[0036]

[0037] The resulting system of equations is solved according to Li's method by using the correlation of the variables of the system of equations with the sample parameters of interest (e.g. layer thickness) assumed according to the model of the sample and by varying the parameters within the scope of the numerical error minimization method until the most likely solution to the system is obtained.

[0038] The known methods yield usable results in the special case of known sample geometries, but cannot be used as a general assumption for determining arbitrary sample geometries, in particular with unknown details.

[0039] From Tsuru, T.: "Tilt-ellipsometry of object surface by specular reflection for three-dimensional shape measurement", Optics Express, Vol. 21, 2013, No. 5, pp. 6625-6632. DOI: 10.1364 / OE.21.006625, there is known a method for 3D determination of reflective samples, in which the sample is illuminated with circularly polarized light. From the shape of the polarization ellipse of the reflected light, conclusions should then be drawn about the position of the surface normal at the illuminated location. Summary of the invention

[0040] The object of the present invention is to improve the known method in such a way that any desired samples, in particular samples with unknown geometries, can be measured using ellipsometry.

[0041] This object is achieved in conjunction with the method for ellipsometric determination of sample properties mentioned above in the technical field in that when formulating the system of equations, the output rotation angle and the input rotation angle are assumed to be independent parameters.

[0042] Preferred embodiments will be further described below.

[0043] The basic idea of ​​the present invention, although it is reasonable based on geometrical aspects, is to get rid of the Li dogma that the input and output rotation angles are equal in magnitude and apply these parameters as independent parameters in the system of equations. The input rotation angle is called γ here, and the output rotation angle is called -δ. In other words, the assumption followed by the present invention is

[0044]

[0045] This has surprising consequences.

[0046] When the system of equations is solved based on a model-based error minimization method similar to Li's method, the additional degrees of freedom now lead to significantly improved agreement of the measured values ​​with the true values ​​outside of those sample geometries for which Li's method is also applicable.

[0047] Thus, within the scope of the imaging variant of the system according to the invention, maps can be created by solving a system of equations pixel by pixel, which maps assign the value of the parameter of interest to each sampling point.

[0048] Thus, in its application similar to the Li method, the method according to the invention already achieves clear advantages. However, the proposal according to the invention is also the basis for a wide range of embodiments which are completely unknown in the prior art.

[0049] Therefore, in a preferred embodiment of the present invention, it is provided that the Mueller parameters N, S, C and / or the ellipsometric transfer parameters Δ and Ψ equivalent thereto are explicitly calculated within the scope of solving a system of equations. This is easy to achieve because the established system of equations is usually overdetermined, or at least non-underdetermined. For the parameters N, S, C, γ, δ, the system of equations contains five unknowns, but can include up to 16 equations depending on the completeness of the measured Mueller matrix. Even without a clear understanding of the proposed rotation angles γ and δ (in principle, the corresponding a priori knowledge cannot be obtained in the case of unknown sample geometry), the ellipsometric transfer parameters Δ and Ψ, which are typically located in the center point of interest in the ellipsometer, can be explicitly calculated. In this regard, the term "solving a system of equations" should be understood in a broad sense and includes all methods known to those skilled in the art, and, if necessary, also includes unknown methods for numerically or analytically finding mathematically direct or indirect solutions to the system of equations.

[0050] Therefore, within the scope of the imaging variant of the system according to the invention, maps can be created by solving a system of equations pixel by pixel, which maps assign a Δ or Ψ value to each sampling point.

[0051] From the calculated ellipsometric transfer parameters Δ and Ψ, as provided in the context of a preferred improvement of the invention, the optical and / or geometrical sample properties can be determined by numerical error minimization as the parameters of the sample model are varied. This corresponds to the usual use of the ellipsometric transfer parameters Δ and Ψ (determined in any way). However, by means of the provision according to the invention, such optical and / or geometrical sample properties can now also be determined for curved or inclined surfaces with high spatial resolution and very short measurement durations.

[0052] In a preferred improvement of this embodiment, it is provided that, within the scope of the error minimization method, the angle of incidence of the illumination light beam onto the surface of the measurement area is used as an additional parameter to be varied. The background can be explained as follows: When calculating the values ​​of Δ and Ψ, it should be noted that these values ​​are usually different from the values ​​measured when the sample is correctly oriented. The reason for this is that the values ​​of Δ and Ψ are usually functions of the angle of incidence of the incident light beam. However, according to the definition, the inclination of the sample will cause the angle of incidence to change compared to the correctly oriented sample, so that usually the values ​​of Δ and Ψ will also deviate from the correctly oriented situation due to the surface inclination. In a well-calibrated ellipsometer, the angle of incidence is known with an accuracy that varies from device to device in the case of a correctly oriented sample. The values ​​Δ and Ψ are usually further used within the scope of the above-mentioned model-based error minimization method to obtain physical properties of the sample such as layer thickness or refractive index, which will be quantitatively included in the calculation result. When the values ​​Δ and Ψ are determined using the method according to the invention, it is therefore advantageous for further evaluation if the deviation from the nominal angle of incidence which has already been set during the measurement on the ellipsometer is also determined as a variable parameter evaluated from the data themselves (or alternatively additionally determined by any other method).

[0053] As described, the (unknown) rotation angles γ and -δ introduced by the method according to the invention mainly represent only auxiliary parameters in the evaluation technology, and quantitative knowledge of these auxiliary parameters is not required for the knowledge of the traditional ellipsometric transfer parameters Δ and Ψ that are in the focus of attention. Nevertheless, in an advantageous variant that can only be realized by the present invention, it is still provided that the input rotation angle γ and the output rotation angle -δ are explicitly calculated within the scope of solving the system of equations. The mathematical possibility for this is caused by the over-determined or non-under-determined nature of the system of equations proposed according to the present invention. The clear knowledge of the rotation angles γ and -δ allows the orientation of the surface normal of the sample surface in the measurement area relative to the reference axis of the ellipsometer to be calculated from them. Therefore, within the scope of the imaging method variant, it is possible to know the position of the surface normal of each sampling point and thus the overall topological structure of the sample surface. However, conversion is not easy.

[0054] The rotation angles γ and -δ calculated by solving the system of equations contain information about the relative orientation of the measured sample surface with respect to the known orientation of the sample normal in the case of a positive sample orientation, when the following angle limits are assumed:

[0055]

[0056] δ=sign(cosφ)cos -1 (cosδ) (8)where

[0057]

[0058] Here, is the nominal incident angle (i.e. the incident angle set on the ellipsometer), is the effective angle of incidence (equal to the actual angle of incidence), and Θ and Φ are the polar and azimuthal angles of the surface normal of the sample.

[0059] Effective angle of incidence is also a function of Θ and Φ, namely:

[0060]

[0061] Therefore, using equations (7) to (10), we can get The values ​​of γ and δ are used to calculate the Mueller matrix rotation angles.

[0062] Equation (9) represents the approximation of the Mueller matrix rotation angle δ on the output side (hereinafter referred to as "approximation model 1"). Strictly speaking, this approximation is that the rotation matrix proposed according to equation (4) cannot be the above output transformation matrix The exact mathematical solution of . However, the validity of the approximation according to equation (9) can be confirmed by experiments. Nevertheless, the following experimental situation can still be imagined, in which other approximate models of the Mueller matrix rotation angle δ on the output side can achieve a more accurate calculation than approximate model 1.

[0063] Therefore, another approximate model (hereinafter referred to as “Approximate Model 2”) was developed, which demonstrates the rotation angle in a different geometric way and provides the following equation for the Mueller matrix rotation angle δ on the output side:

[0064]

[0065] It has the variable name defined above.

[0066] Therefore, using equations (7), (8), (10), and (11), we can get from Θ and Φ and The values ​​of γ and δ are used to calculate the Mueller matrix rotation angles.

[0067] On the contrary, if γ and δ are already known at the time of measurement (because they are set) If θ is determined, then the above equation (7) and optionally equation (9) or (11) form a system of equations consisting of two equations with unknowns θ and Φ. By solving this system of equations, the polar angle and azimuth of the surface normal can be determined, and thus the orientation of the surface normal relative to the laboratory coordinate system can be determined. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] Further details and advantages of the invention emerge from the following detailed description and the accompanying drawings.

[0069] in:

[0070] Figure 1 : A schematic diagram showing the basic structure of an ellipsometer;

[0071] Figure 2 : shows a schematic diagram for illustrating a sample properly oriented in a conventional manner;

[0072] Figure 3 : A schematic diagram showing a sample for illustrating a tilted orientation;

[0073] Figure 4 : A schematic diagram for explaining the input rotation angle γ is shown,

[0074] Figure 5 : A schematic diagram for explaining the output rotation angle δ according to the approximate model 1 is shown;

[0075] Figure 6 : A schematic diagram for explaining the output rotation angle δ according to approximate model 2 is shown. DETAILED DESCRIPTION

[0076] Figures 1 to 5 It is intended to be used to illustrate and check the geometric plausibility of relevant parameters in the context of the present invention.

[0077] Figure 1 The basic structure of an ellipsometer 10 is shown, as is known from the prior art and which can also be used within the scope of the present invention. 入 (illumination beam) illuminates the sample 12 with a surface normal n. Therefore, the incident input beam k 入 Here the angle of incidence relative to the sample normal n is With the aid of the polarizer 18 and the downstream compensator 20, the incident input beam k 入 After reflection on the surface of the sample 12, the light is used as the probe beam k 出 The output beam k is supplied to the detector 24. The polarization state of the light is changed by reflection on the sample surface in a manner that is characteristic of the physical properties of the sample, such as coating thickness and / or optical refractive index. 出 Before the detector 24 , the (probe beam) is guided through an analyzer 26 , the variable setting of which influences the polarization-dependent intensity reaching the detector 24 .

[0078] Figure 1 The illustration of shows a non-imaging ellipsometer 10. An imaging ellipsometer differs therefrom only in the imaging optics between the sample 12 and the analyzer 26 or optionally between the analyzer 26 and the detector 24 and by the detector 24 forming the image.

[0079] Figure 2 In a more schematic way, the geometry inside the ellipsometer 10 is shown when the sample 12 is oriented "correctly" in the usual way. The reference axis of the ellipsometer 10, denoted here by z, coincides with the sample normal n. At the same time, the input beam k 入 and the output beam k 出 In this orientation of the sample, the input beam k 入 The actual angle of incidence relative to the sample normal n With input beam k 入 The nominal angle of incidence relative to the reference axis z of the ellipsometer 10 is same.

[0080] This situation is similar to Figure 3 The tilted orientation of sample 12 depicted in FIG. Figure 3 The illustration can also be applied to curved sample surfaces, as long as each curved surface can be considered as a network of very small inclined surfaces. In the case of a tilted sample surface, the actual angle of incidence is With the nominal angle of incidence Differently, the sample normal n no longer coincides with the reference axis z of the ellipsometer 10. The orientation of the sample normal n can be described in the eigensystem of the ellipsometer 10 by a polar angle Φ and an azimuthal angle Θ.

[0081] Figure 4 is a sketch for explaining and checking the rationality of the input rotation angle γ discussed in the context of the foregoing description of the present invention. γ corresponds to the actual incident plane (i.e., the plane formed by the input beam k 入 and the actual output beam k 出 The angle at which the developed plane is tilted relative to the nominal plane of incidence, i.e. the yz plane (calibration plane) of the ellipsometer, is perpendicular to the input beam k. 入 Measured in the plane of .

[0082] Figure 5 It is used to illustrate and check the rationality of the output rotation angle δ according to the approximate model 1 discussed in the above description. δ is equivalent to the angle at which the "virtual emission plane" defined below is tilted relative to the nominal emission plane when the sample is correctly oriented, which is perpendicular to the nominal output beam k 出,0 The “virtual exit plane” corresponds to the plane formed by the nominal exit beam k 出,0 and when the sample normal is in the actual outgoing beam k 出 Three-dimensional rotation to the nominal outgoing beam k 出,0 The above is expanded by the vector obtained from the sample normal n when it is hypothetically moved as a rigid body.

[0083] Figure 6 It is used to illustrate and check the rationality of the output rotation angle δ according to the approximate model 2 discussed in the above description. δ is equivalent to the angle between the auxiliary vector defined below and the nominal emission plane, that is, the nominal emission plane when the sample is correctly oriented, which is perpendicular to the nominal output beam k 出,0 The auxiliary vector mentioned here is measured in the plane perpendicular to the output beam k 出 and the vector lying in the actual emission plane passes through the nominal output beam k 出,0 The direction of δ is obtained by projecting it onto the mentioned plane in which δ is measured.

[0084] Reference numerals list

[0085] 10 Ellipsometer

[0086] 12 samples

[0087] 14 Light Source

[0088] 18 Polarizer

[0089] 20 Analyzer

[0090] 24 Detector

[0091] k 入 Input beam

[0092] k 出 Output beam

[0093] k 出,0 Nominal output beam

[0094] n Sample normal

[0095] Angle of incidence

[0096] Nominal angle of incidence The actual angle of incidence Φ Polar angle

[0097] θ Azimuth

[0098] γ Input rotation angle -δ Output rotation angle

[0099] x, y, z Cartesian coordinates of the ellipsometer eigensystem

Claims

1. A method for determining the properties of a sample (12) by ellipsometry, the method comprising the following steps: - positioning the sample (12) in an ellipsometer (10), wherein a surface normal (n) of a measurement area of ​​the sample surface is inclined relative to a reference axis (z) of the ellipsometer (10), - measuring a Mueller matrix for said measurement region, - Establishing an at least non-underdetermined system of equations based on equating the matrix product of the measured Mueller matrix with the following matrix: + the rotation matrix about the input rotation angle (γ), + the isotropic Mueller matrix in normalized NCS form, and +Rotation matrix about the output rotation angle (-δ), - solving said system of equations according to the parameters representing the characteristics of the sample to be determined, It is characterized in that When establishing the equation group, the input rotation angle (γ) and the output rotation angle (-δ) are assumed to be independent parameters.

2. The method according to claim 1, It is characterized in that Within the scope of solving the system of equations, the Mueller parameters N, S, C and / or the ellipsometric transfer quantities Δ and Ψ equivalent to these Mueller parameters are explicitly calculated.

3. The method according to claim 2, The method further comprises the following steps: - The optical and / or mechanical properties of the sample are obtained from the calculated ellipsometric transfer parameters Δ and Ψ by numerical error minimization as the parameters of the sample model are varied.

4. The method according to claim 3, It is characterized in that In the context of the error minimization method, the incident angle Used as additional parameters to be varied.

5. The method according to claim 1, Within the scope of solving the system of equations, the input rotation angle (γ) and the output rotation angle (−δ) are explicitly calculated.

6. The method according to claim 5, It is characterized in that The orientation of the surface normal (n) of the sample surface in the measurement area relative to the reference axis (z) of the ellipsometer (10) is calculated from the rotation angles (γ, −δ) known by solving the system of equations.

7. The method according to claim 1, It is characterized in that As the parameters of the sample model are varied, the system of equations is solved by numerical error minimization.

8. The method according to any one of claims 1 to 7, It is characterized in that The ellipsometer (10) is an imaging ellipsometer (10).

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